Phy 211 General Physics I Chapter 3 Vectors Lecture Notes Vectors Scalars Most physical quantities can categorized as one of 2 types tensors notwithstanding 1 Scalar Quantities described by a single number a unit s Example the length of the driveway is 3 5 m 2 Vectors Quantities described by a value magnitude direction Example the wind is blowing 20 m s due north Vectors are represented by an arrow where 1 the length of the arrow is proportional to the magnitude of the vector A 2 m B 2 A 4m 2 The direction of the arrow represents the direction of the vector 1 2 3 4 Properties of Vectors Only vectors of the same kind can be added together 2 or more vectors can be added together to obtain a resultant vector The resultant vector represents the combined effects of multiple vectors acting on the same object system Direction as well as magnitude must be taken into account when adding vectors When vectors are co linear they can be added like scalars A B A B R Any single vector can be treated as a resultant vector and represented as 2 or more component vectors A Ax 5 R Ay Ay Ax To add vectors of this type requires sophisticated mathematics or use of graphical techniques Trigonometry Review remember SOHCAHTOA 1 The relationships between the sides and angles of right triangles are well defined 2 Consider the following right triangle C A Three primary trig relations relative to Sine of sin opposite hypotenuse A C Cosine of cos adjacent hypotenuse B C Adding Vectors or otherwise A Graphic Method To add 2 vectors place them tail to head without changing their direction the sum resultant is the vector obtained by connecting the tail of the first vector with the head of the second vector a R A B means the vector R is the sum of vectors A and B b Note R A B the magnitude of the vector R is NOT necessarily equal to the sum of the magnitudes of vectors A and B In general 2 2 2 R A B 2 A B cos R A B Other Notes 1 For co linear vectors pointing in the same direction R A B 2 For co linear vectors pointing in opposite directions R A B Vector Addition cont B Component Method 1 Express each vector as the sum of 2 component vectors The direction of each component vector should be the same for both vectors It is common to use the horizontal and vertical directions These vectors are the horizontal and vertical components of the vector Example vector A vector B A A A Ax horizontal and Ay vertical or x y A x i A y j Bx horizontal and By vertical or B Bx By Bx i By j Note The unit vectors i and j indicate the directions of the vector components 2 The magnitudes for corresponding component vectors for A B can now be added together like scalars to obtain the component vectors for the resultant vector Rx Ax Bx and Ry Ay By R R x R y R x i R y j And thus The magnitude of the resultant is then component vectors by using obtained from the the Pythagorean Theorem 2 2 R Rx Ry Rx Ry 3 To calculate the components we need to know the magnitude R of the vector and the y angle a it makes with the horizontal direction cos Rx R since Rx Rcos sin Ry R since Ry Rsin R Ry Rx x The Scalar Dot Product Two vectors A and B can be multiplied to product a scalar resultant called the scalar or Dot product When using the magnitudes of the vectors A iB A B cos where is the angle between vectors A and B When using vector components A iB A x Bx A y By Useful properties of scalar products A B B A A A A2 i i j j k k 1 Example The scalar product of the vectors of force and displacement is used to calculate work performed by the force The Vector Cross Product 1 2 3 4 Two vectors A and B can be multiplied to produce a vector resultant called the vector or cross product When using the magnitudes of the vectors A B A B sin where is the angle between vectors A and B The direction of the vector product is perpendicular to the plane of the vectors A B When using vector components A B A y Bz A z By i A z Bx A x Bz j A x By A y Bx k Notes a The presence of the vector product implies that 3 spatial dimensions are specified b The vector product is perpendicular to both A and B J Willard Gibbs 1839 1903 Considered one of the greatest scientists of the 19th century Major contributions in the fields of Thermodynamics Statistical mechanics Formulated a concept of thermodynamic equilibrium of a system in terms of energy and entropy Chemistry Chemical equilibrium and equilibria between phases I m sure you ve heard of the Gibb s Free Energy Mathematics Developed the foundation of vector mathematics Physics Humor 1 What do you get when you cross an apple with a grape Ans Apple Grape sin 2 What do you get when you cross an apple with a alligator Ans Nothing alligators are scalar
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